New Model of massive spin-2 particle Based on Phys.Rev. D90 (2014) 043006, Y.O, S. Akagi, S. Nojiri Phys.Rev. D90 (2014) 123013, S. Akagi, Y.O, S. Nojiri Yuichi Ohara QG lab. Nagoya univ.
Introduction Massive spin-2 = Massive graviton? The free massive spin-2 field theory was formulated by Fierz and Pauli. (They tried to construct field theories with arbitrary spin) No ghost (Consistent theory as QFT) Realization of 5 d.o.f in 4 dimensions (Massive spin-2 particle) thanks to the Fierz-Pauli mass term. ? Massive spin-2 particle Massive graviton
Introduction The 1 st problem : vDVZ discontinuity Massive spin-2 in Linearized GR the massless limit Does this mean the massive spin-2 particle can not be graviton? Vainshtein’s argument Non-linearity screens the discontinuity!
Introduction Einstein-Hilbert + Fierz-Pauli mass term Static and Schwarzschild spherical solution Solution of EH + FP 𝑛 → 0 Fully the non-linear Fully the non-linear massive spin-2 massless spin-2 No discrepancy!
Introduction Full nonlinearity Fierz-Pauli mass term (Gravity) (Ghost-free) Massive spin-2 = Massive graviton ? However… Boulware and Deser suggested the nonlinearity and the ghost- free property are not compatible with each other.
Introduction The 2 nd problem : Boulware-Deser ghost ghost Nonlinearity e.g.) Einstein-Hilbert + Fierz-Pauli mass term ( ℎ 𝑗𝑘 ≔ ij − 𝜀 𝑗𝑘 ) ADM variables ( Lapse 𝑂 , shift 𝑂 𝑗 , 3-metric 𝛿 𝑗𝑘 ) Hamiltonian constraint Momentum constraints
Introduction Progress in 2000s 1. DGP model Phys.Lett. B485 (2000) 208-214 Higher derivative scalar field theory without any ghost. 2. Effective field theoretical approach Annals Phys. 305 (2003) 96-118 Stuckelberg trick Encoding the scalar mode into the lagrangian explicitly. (Using the scalar field) The origin of the Boulware-Deser ghost is the higher derivative of the scalar field.
Introduction Progress in 2000s Field theoretical approach DGP model (Ghost-free massive gravity) (Stuckelberg method) The origin of BD ghost : Higher Higher derivative scalar derivatives of the scalar field. field theory without ghost. dRGT massive gravity
Introduction dRGT massive gravity de Rham, Gabadadze, Tolley Phys.Rev.Lett. 106 (2011) 231101 Nonlinearity and the ghost-free property are compatible now!
Introduction Full nonlinearity Potential terms (Gravity) (Ghost-free) Massive spin-2 = Massive graviton (dRGT massive gravity) Massive spin-2 particles can be identified with massive gravitons. Should we identify the massive spin-2 with the massive graviton?
Motivation Question 1 Is Massive spin-2 = Massive graviton necessary? Massive spin-2 theory necessarily leads to modification of gravity? As a fact, There exist massive spin-2 particles in the hadron spectrum.
Motivation Question 2 Which assumptions can we remove? In the history of the massive spin-2 field…. Vainshten mechanism vDVZ discontinuity Full non-linearity (EH term is introduced) This is natural in some sense because…. To avoid the vDVZ discontinuity. The spin-2 field ℎ μ𝜉 is naturally replaced by the metric 𝜈𝜉
Motivation Question 2 Which assumptions can we remove? Einstein-Hilbert term The massive spin-2 particle is not the graviton in this point of view. Full nonlinearity is not necessary. Construct the massive spin-2 theory.
Fierz-Pauli theory Massless spin-2 field theory The phase space is spanned by ℎ 𝑗𝑘 and 𝜌 𝑗𝑘 . (12 dimensions) 4 first class constraints 8 second class constraints 4 gauge fixing functions (12 dimensional phase space) − (8 constraints) = 4 independent comp. Massless spin-2 particle has 2 degrees of freedom.
Fierz-Pauli theory Massive spin-2 field theory Possible quadratic terms Candidates for mass terms When 𝑏 ≠ 0 , an extra d.o.f propagates with a negative kinetic energy. Fierz-Pauli lagrangian Fierz-Pauli tuning
Fierz-Pauli theory Hamiltonian analysis Conjugate momenta Lagrangian density
Fierz-Pauli theory ℎ 00 : Lagrange multiplier (Linear) → Single constraint Secondary constraint
Fierz-Pauli theory In total, we have two second class constraints. (12 dimensional phase space) − (2 constraints) = 10 independent comp. (5 polarizations of the massive spin-2 particle) 2 does not appear thanks to the Fierz-Pauli tuning. ℎ 00 No ghost if ℎ 00 remains linear in general.
Ghost-free interaction Ghost-free interactions for Fierz-Pauli theory Folkerts et al. arXiv:1107.3157 [hep-th] Ghost-free term Hinterbichler, JHEP 10 (2013) 102 d : The number of derivatives, n : The number of the fields, D : Spacetime dim The kinetic term and the mass term are included. • We use this term to construct the massive spin-2 model. •
Ghost-free interaction Linear with respect to ℎ 00 in the Hamiltonian. The terms which include both of ℎ 00 and ℎ 0𝑗 never appear. Variation of ℎ 00 a constraint for ℎ 𝑗𝑘 and their conjugate momenta 𝜌 𝑗𝑘 + secondary constraint (12 dimensional phase space) − (2 constraints) = 10 independent comp. No ghost
Ghost-free interaction The Fierz-Pauli lagrangian The kinetic term : The mass term :
Ghost-free interaction In 4 dimensions, the allowed interaction is following: Non-derivative int. Derivative int. Other possibilities are excluded due to the antisymmetric properties.
New model of massive spin-2 New model of massive spin-2 𝜈 , 𝜂 , 𝜇 : constants
New model of massive spin-2 Possible application (Additional motivation) Supersymmetry breaking mechanism? Can this model be used to realize SUSY breaking? BH physics and cosmology? The new spin-2 model on curved spacetime.
FP theory in curved space The simplest model (Minimal coupling) We don’t regard the massive spin -2 as the perturbation of metric. Unfortunately, this model does not have 5 degrees of freedom. To see this reason, let us see the FP theory in flat spacetime.
FP theory in curved space Taking the variation gives e.o.m Two constrains obtained from 𝐹 μ𝜉 , ,
FP theory in curved space Key point Commutativity of 𝜖 μ Existence of the second equation On the other hand…… Covariant derivatives 𝛼 μ do not commute with each other. type terms appear and the constraint is lost.
FP theory in curved space FP theory in curved spacetime was considered by Buchbinder et al . I. L. Buchbinder, D. M. Gitman, V. A. Krykhtin and V. D. Pershin, Nucl. Phys. B 584 (2000) 615 [hep-th/9910188] I. L. Buchbinder, V. A. Krykhtin and V. D. Pershin, Phys. Lett. B 466 (1999) 216 [hep-th/9908028]. They constructed the theory having 5 d.o.f in curved spacetime. Problem : type terms appear and the constraint is lost. Prepare non-minimal coupling terms like (quadratic in derivatives)
FP theory in curved space Prepare non-minimal coupling terms like (quadratic in derivatives) They determined 𝑏 𝑗 and found that the theory can be ghost-free on Einstein manifold.
FP theory in curved space Ghost-free FP theory on curved space 𝜊 : Real parameter The background is restricted to Einstein manifold
New model of massive spin-2 in a curved spacetime Interaction on the Einstein manifold Int. in a flat spacetime Int. on Einstein manifold
New model of massive spin-2 in a curved spacetime New model of massive spin-2 on the Einstein manifold Is this model ghost-free on Einstein manifold? Counting the degrees of freedom using Lagrangian analysis.
New model of massive spin-2 in a curved spacetime Lagrangian analysis 1. The system containing some set of fields 𝜚 𝐵 x , A = 1, 2, ⋯ 𝑂 2. The second time derivatives are defined only for 𝑠 < 𝑂 fields in e.o.m. 3. N-r primary constraints are constructed from e.o.m. 4. Requirement of conservation in time of the primary constraints defines the second time derivatives for remaining fields or new secondary constraints. 5. This procedure continues until the second time derivatives are defined for all fields ϕ 𝐵
New model of massive spin-2 in a curved spacetime Example : FP theory in a flat spacetime 1. The system containing some set of fields 𝜚 𝐵 x , A = 1, 2, ⋯ 𝑂 ℎ μ𝜉 : 10 components Equations of motion
New model of massive spin-2 in a curved spacetime 2. The second time derivatives are defined only for 𝑠 < 𝑂 fields in e.o.m. ℎ 𝑗𝑘 : 6 components 3. N-r primary constraints are constructed from e.o.m. 4 constraints (@ some time 𝑢 ) : Undetermined (4 components)
New model of massive spin-2 in a curved spacetime 4-1. Requirement of conservation in time of the primary constraints . This equations do not contain and are eliminated using e.o.m. Secondary constraint-1 𝜚 1 𝜈 = 0 , (some time) (all time) Continue the same procedure.
New model of massive spin-2 in a curved spacetime 4-2. Requirement of conservation in time of the primary constraints . This equations do contain and determine the dynamics of On the other hand, This equations do not contain any time derivative of h. Secondary constraint-2 (all time) (some time) Constraints
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